Exercises on reason and proportion

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In mathematics, when we want to compare two quantities, we calculate the quotient between their respective measurements. This quotient is called reason.

The equality between two reasons is called proportion and, according to the ratio of variation between the quantities, we can have quantities directly or inversely proportional.

Directly proportional quantities: when an increase in one of them leads to an increase in the other, or a reduction in one leads to a reduction in the other.
Indirectly proportional quantities: when the increase of one of them leads to the reduction of the other, or when the reduction of one of them leads to the increase of the other.

To learn more, check out a list of solved exercises about ratio and proportion, which we prepared.

Index

List of exercises on ratio and proportion
Resolution of question 1
Resolution of question 2
Resolution of question 3
Resolution of question 4
Resolution of question 5
Resolution of question 6
Resolution of question 7
Resolution of question 8

List of exercises on ratio and proportion

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Question 1. Determine the ratio between the area of a square with sides equal to 50 centimeters and a square with sides equal to 1.5 meters. Interpret the number obtained.

Question 2. In a math test with 15 questions, Eduarda got 12. What was Eduarda's performance on the test?

Question 3. The distance between two cities is 180 kilometers, but on a map, this distance was represented by 9 cm. What scale is used on this map? Interpret the scale obtained.

Question 4. Check if the reasons below form a ratio:

The) $\dpi{100} \bg_white \large \frac{3}{8} \: \mathrm{e }\: \frac{9}{24}$

B) $\dpi{100} \bg_white \large \frac{2}{5} \: \mathrm{e }\: \frac{18}{25}$

ç) $\dpi{100} \bg_white \large \frac{150}{50} \: \mathrm{e }\: \frac{12}{4}$

Question 5. Determine the value of $\dpi{100} \bg_white \large x$ in each of the following proportions:

The) $\dpi{100} \bg_white \large \frac{x}{7}= \frac{9}{63}$

B) $\dpi{100} \bg_white \large \frac{8}{32}= \frac{2}{x}$

ç) $\dpi{100} \bg_white \large \frac{2}{10}= \frac{3}{2x}$

d) $\dpi{100} \bg_white \large \frac{3,7}{11}= \frac{x}{55}$

and) $\dpi{100} \bg_white \large \frac{2}{9}= \frac{x + 8}{x + 50}$

Question 6. Determine the value of $\dpi{100} \bg_white \large x$ in the following proportion:

$\dpi{100} \bg_white \large \frac{x}{6} = \frac{24}{x}$

Question 7. To make a bread recipe, 3 eggs are needed for every 750 grams of wheat flour. How many eggs will be needed for 5 kg of flour.

Question 8. To finish a job, 15 workers spend 30 days. How many days did 9 workers spend to finish this same work?

Resolution of question 1

We have a square with a side equal to 50 cm and a square with a side equal to 1.5 m.

We need the measurements in the same unit. So, let's transform 1.5 m to centimeters:

1.5 x 100 cm = 150 cm

That is, 1.5 m = 150 cm.

Now let's calculate the area of each of the squares:

THE one square area is given by the measure of the squared side:

L = 50 cm ⇒ Area = 2500 cm ²

L = 150 cm ⇒ Area = 22500 cm ²

Thus, the ratio between the area of the square with the side equal to 50 cm and the area of the square with the side equal to 150 cm is given by:

$\dpi{100} \bg_white \large Raz\tilde{a}o = \frac{2500}{22500} = \frac{1}{9}$

Interpretation: The area of the square with a side equal to 1.5 m is 9 times the area of the square with a side equal to 50 cm.

Resolution of question 2

Let's calculate the ratio between the number of questions Eduarda got right and the number of questions in the test:

$\dpi{100} \bg_white \large Raz\tilde{a}o = \frac{12}{15} = \frac{4}{5}$

This ratio means that for every 5 questions, Eduarda got 4 right and as 4/5 = 0.8, so Eduarda's use in the test was 80%.

Resolution of question 3

Scale is a special type of ratio between the length in the drawing and the actual length.

We have:

Distance on map = 9 cm

Actual distance = 180 km

First, we must express both measures in the same unit. Let's transform 180 km to centimeters:

180 x 100000 cm = 180 00000 cm

Thus, 180 km = 180 00000 cm.

Now, let's calculate the scale:

$\dpi{100} \bg_white \large Scale = \frac{9}{18000000} = \frac{1}{2000000}$

Interpretation: The scale used on the map was 1: 2000000, this means that 1 cm on the map corresponds to 2000000 cm in actual distance.

Resolution of question 4

A proportion is an equality between two ratios and one of the properties of a proportion is that the product of the extreme terms is equal to the product of the middle terms.

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Thus, to find out if two ratios form a proportion, just multiply crossed and check if the result obtained is the same.

The) $\dpi{100} \bg_white \large \frac{3}{8} \: \mathrm{e }\: \frac{9}{24}$

3. 24 = 72

9. 8 = 72

The result is the same for both products, so the ratios form a ratio.

B) $\dpi{100} \bg_white \large \frac{2}{5} \: \mathrm{e }\: \frac{18}{25}$

2. 25 = 50

18. 5 = 90

The result is not the same for both products, so the ratios do not form a ratio.

ç) $\dpi{100} \bg_white \large \frac{150}{50} \: \mathrm{e }\: \frac{12}{4}$

150. 4 = 600

12. 50 = 600

The result is the same for both products, so the ratios form a ratio.

Resolution of question 5

To determine the value of x, simply multiply cross and solve the corresponding equation.

The) $\dpi{100} \bg_white \large \frac{x}{7}= \frac{9}{63}$

$\dpi{100} \bg_white \large 63\cdot x = 7 \cdot 9\Rightarrow 63\cdot x = 63 \Rightarrow x = \frac{63}{63} \Rightarrow x = 1$

B) $\dpi{100} \bg_white \large \frac{8}{32}= \frac{2}{x}$

$\dpi{100} \bg_white \large 8\cdot x = 2 \cdot 32\Rightarrow 8\cdot x = 64 \Rightarrow x = \frac{64}{8} \Rightarrow x = 8$

ç) $\dpi{100} \bg_white \large \frac{2}{10}= \frac{3}{2x}$

$\dpi{100} \bg_white \large 2\cdot 2x = 3 \cdot 10\Rightarrow 4\cdot x = 30\Rightarrow x = \frac{30}{4} \Rightarrow x = 7.5$

d) $\dpi{100} \bg_white \large \frac{3,7}{11}= \frac{x}{55}$

$\dpi{100} \bg_white \large 11\cdot x = 3.7 \cdot55\Rightarrow 11\cdot x = 203.5 \Rightarrow x = \frac{203.5}{11} \Rightarrow x = 18.5$

and) $\dpi{100} \bg_white \large \frac{2}{9}= \frac{x + 8}{x + 50}$

$\dpi{100} \large 2\cdot (x + 50) = 9 \cdot (x + 8)\Rightarrow 2x + 100 = 9x + 72x$

$\dpi{100} \bg_white \large \Rightarrow 7x = 28 \Rightarrow x = \frac{28}{7} \Rightarrow x = 4$

Resolution of question 6

$\dpi{100} \bg_white \large \frac{x}{6} = \frac{24}{x}$

Multiplying cross, we get:

$\dpi{100} \bg_white \large x\cdot x = 24 \cdot 6\Rightarrow x^2 = 144\Rightarrow x = \sqrt{144} \Rightarrow x = \pm 12$

Resolution of question 7

First, let's write the two flour measurements in the same unit. Let's transform 5 kg to grams:

5 x 1000 grams = 5000 grams

So 5 kg = 5000 grams.

We have a proportion with an unknown value:

3 eggs → 750 grams of flour

x eggs → 5000 grams of flour

I.e,

$\dpi{100} \bg_white \large \frac{3}{x} = \frac{750}{5000}$

Let's multiply cross to find the value of x:

$\dpi{100} \bg_white \large 750\cdot x = 3\cdot 5000\Rightarrow 750 \cdot x = 15000\Rightarrow x = \frac{15000}{750} \Rightarrow x = 20$

So, for 5 kg of wheat flour, 20 eggs will be needed.

Resolution of question 8

We have a proportion with an unknown value:

15 workers → 30 days

9 workers → x days

Note that when the number of workers decreases, the number of days to complete the work must increase. Thus, the ratios are indirectly proportional and we must change the order of the numerator and denominator of one of them:

$\dpi{100} \bg_white \large \frac{15}{9} = \frac{x}{30}$